ABSTRACT

In this paper, by utilizing the fractional calculus theory and computer simulations, dynamics of the fractional order system is studied. Further, we have extended the nonlinear feedback control in ODE systems to fractional order systems, in order to eliminate the chaotic behavior. The results are proved analytically by stability condition for fractional order system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.

1. Introduction

Fractional calculus is a classical mathematical concept, with a history as long as calculus itself. It is a generalization of ordinary differentiation and integration to arbitrary order, and is the fundamental theories of fractional order dynamical systems. Fractional-order differential/integral has been applied in physics and engineering, such as viscoelastic system [1], dielectric polarization [2], electrode-electrolyte polarization [3] and electromagnetic wave [4], and so on.

The fractional order system and its potential application in engineering field become promising and attractive due to the development of the fractional order calculus. Typically, chaotic systems remain chaotic when their equations become fractional. For example, it has been shown that the fractional order Chua’s circuit with an appropriate cubic nonlinearity and with an order as low as 2.7 can produce a chaotic attractor [5].

However, there are essential differences between ordinary differential equation systems and fractional order differential systems. Most properties and conclusions of ordinary differential equation systems cannot be extended to that of the fractional order differential systems. Therefore, the fractional order systems have been paid more attention. Recently, many investigations were devoted to the chaotic dynamics and chaotic control of fractional order systems [6-12].

In this paper, practical scheme is proposed to eliminate the chaotic behaviors in fractional order system by extending the nonlinear feedback control in ODE systems to fractional-order systems. This paper is organized as follows. In Section 2, the numerical algorithm for the fractional order system is briefly introduced. In Section 3, Dynamics of the fractional order system is numerically studied. In section 4, general approach to feedback control scheme is given, and then we have extended this control scheme to fractional order system, numerical results are shown. Finally, in Section 5, concluding comments are given.

2. Fractional Derivative and Numerical Algorithm

There are two approximation methods for solving fractional differential equations. The first one is an improved version of the Adams-Bashforth-Moulton algorithm, and the rest one is the frequency domain approximation. The Caputo derivative definition involves a time-domain computation in which nonhomogenous initial conditions are needed, and those values are readily determined. In this paper, the Caputo fractional derivative defined in [13] is often described by

when is the first integer that is not less than, is the α-order Riemann-Liouville integral operator which defined by

where is the Gamma function,

Now we consider the fractional order system [14] which is given by

(1)

where is the fractional order,

By exploiting the Adams-Bashforth-Moulton scheme [15], the fractional order system (1) can be discretized as followings:

3. Dynamic Analysis of the Fractional Order System

Theorem 1: The fractional linear autonomous system

is locally asymptotically stable if and only if

Theorem 2: Suppose be an equilibrium point of a fractional nonlinear system

If the eigenvalues of the Jacobian matrix satisfy

then the system is locally asymptotically stable at the equilibrium point

The system (1) has five equilibrium points:

where

When we obtain

First, we choose to study, the eigenvalues of the Jacobian matrix are and We can obtain and According to Theorem 2, we can easily conclude that the equilibrium of system (1) is unstable when and are all greater than zero.

We choose and to study, the eigenvalues of the Jacobian matrix are and We can obtain and According to Theorem 2, we can easily conclude that when and are all less than the equilibrium of system (1) is stable. On the contrary, when and are all great than, the equilibrium of system (1) is unstable.

Finally , when choose and to study, the eigenvalues of the Jacobian matrix are and We can obtain and According to Theorem 2, we can easily conclude that when and are all great than, the equilibrium of system (1) is unstable.

In sum, there exists at least one stable equilibrium and of system (1), when and are all less than, i.e., the system (1) will be stabilized at one point finally; when and are all greater than, all the equilibriums of system (1) are unstable, the system (1) will exhibit a chaotic behaviour; when the problem will be complicated, the system (1) may be convergent, periodic or chaotic. For example, when the value of the largest Lyapunov exponent is 0.1653. Obviously, the fractional order system (1) is chaotic. When the fractional order system (1) is not chaotic, but periodic orbits appear.

4. Feedback Control

Let us consider the fractional order system

(2)

where is the system state vector, and the control input vector. Given a reference signal the problem is to design a controller in the state feedback form:

where is the vector-valued function, so that the controlled system

can be driven by the feedback control g(x, t) to achieve the goal of target tracking so we must have

Let be a periodic orbit or fixed point of the given system (2) with, then we obtain the system error

where and

Theorem 3: If is a fixed point of the system (2) and the eigenvalues of the Jacobian matrix at the equilibrium point satisfies the condition

then the trajectory of system (2) converge to

Let us consider the fractional order system (2), we propose to stabilize unstable periodic orbit or fixed point, the controlled system is as follows:

(3)

Since is solution of system (1), then we have:

(4)

Subtracting (4) from (3) with notation, we obtain the system error

(5)

We define the control function as follow

(6)

So the system error (5) becomes

(7)

The Jacobian matrix of system (7) is

so we have the eigenvalues and When all eigenvalues are real negatives, one has therefore for all satisfies it follows from Theorem 3 that the trajectory of system (2) converges to and the control is completed.

5. Numerical Simulation

In this section we give numerical results which prove the performance of the proposed scheme. As mentioned in Section 2 we have implemented the improved AdamsBashforth-Moulton algorithm for numerical simulation.

The control can be started at any time according to our needs, so we choose to activate the control when in order to make a comparison between the behavior before activation of control and after it.

For and q₃ = 0.98, unstable point has been stabilized, as shown in Figure 1, note that The control is activated when and the evolution of is chaotic, then when the control is started at we see that is rapidly stabilized.

For and the unstable point has been stabilized, as shown in Figure 2.

For and the unstable point has been stabilized, as shown in Figure 3.

For the unstable point has been stabilized, as shown in Figure 4.

For the unstable point has been stabilized, as shown in Figure 5.

When is less than, there is a chaotic behavior,

but when the control is activated at, the five points and are rapidly stabilized.

6. Conclusions

Chaotic phenomenon makes prediction impossible in the real world; then the deletion of this phenomenon from fractional order system is very useful, the main contribution of this paper is to this end.

In this paper, we investigate the system with fractional order applying the fractional calculus technique. According to the stability theory of the fractional order system, dynamical behaviors of the fractional order system are analyzed, both theoretically and numerically. Furthermore, nonlinear feedback control scheme has been extended to control fractional order system. The results are proved analytically by stability condition for fractional order system. Numerically the unstable fixed points have been successively stabilized for different values of and Numerical results have verified the effectiveness of the proposed scheme.

7. Acknowledgements

This work is supported by the Qing Lan Project of Jiangsu Province under the Grant Nos. 2010 and the 333 Project of Jiangsu Province under the Grant Nos. 2011.

REFERENCES

NOTES

^*Corresponding author.